I'm writing and correcting API documentation in Sphinx using reStructuredText. Does anyone know of a good spellchecker that will handle this format, or a bunch of Unix/Linux tools that will allow me to use Aspell with it?

Since aspell has options for checking HTML and TeX files, you can spell-check some HTML or Tex produced by Sphinx. I imagine you could also use aspell on text output from Sphinx - assuming that removes any mark-up (e.g. asterisks) that might confuse aspell .